Advertisement

Median Trajectories

  • Kevin Buchin
  • Maike Buchin
  • Marc van Kreveld
  • Maarten Löffler
  • Rodrigo I. Silveira
  • Carola Wenk
  • Lionov Wiratma
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6346)

Abstract

We investigate the concept of a median among a set of trajectories. We establish criteria that a “median trajectory” should meet, and present two different methods to construct a median for a set of input trajectories. The first method is very simple, while the second method is more complicated and uses homotopy with respect to sufficiently large faces in the arrangement formed by the trajectories. We give algorithms for both methods, analyze the worst-case running time, and show that under certain assumptions both methods can be implemented efficiently. We empirically compare the output of both methods on randomly generated trajectories, and analyze whether the two methods yield medians that are according to our intuition. Our results suggest that the second method, using homotopy, performs considerably better.

Keywords

Span Tree Time Stamp Dynamic Time Warping Medial Axis Homotopy Type 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agarwal, P.K., de Berg, M., Gao, J., Guibas, L.J.: Staying in the middle: Exact and approximate medians in R1 and R2 for moving points. In: Proc. CCCG, pp. 43–46 (2005)Google Scholar
  2. 2.
    Agarwal, P.K., Gao, J., Guibas, L.J.: Kinetic medians and kd-trees. In: Möhring, R.H., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 5–16. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  3. 3.
    Agarwal, P.K., Guibas, L.J., Hershberger, J., Veach, E.: Maintaining the extent of a moving point set. Discrete Comput. Geom. 26, 353–374 (2001)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Amenta, N., Bern, M.W., Eppstein, D., Teng, S.-H.: Regression depth and center points. Discrete Comput. Geom. 23, 305–323 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Buchin, K., Buchin, M., Gudmundsson, J., Löffler, M., Luo, J.: Detecting commuting patterns by clustering subtrajectories. In: Hong, S.-H., Nagamochi, H., Fukunaga, T. (eds.) ISAAC 2008. LNCS, vol. 5369, pp. 644–655. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Cabello, S., Liu, Y., Mantler, A., Snoeyink, J.: Testing homotopy for paths in the plane. Discrete Comput. Geom. 31, 61–81 (2004)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Chin, F.Y.L., Snoeyink, J., Wang, C.A.: Finding the medial axis of a simple polygon in linear time. Discrete Comput. Geom. 21, 405–420 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Durocher, S., Kirkpatrick, D.: The projection median of a set of points. Comput. Geom. 42, 364–375 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Durocher, S., Kirkpatrick, D.G.: The Steiner centre of a set of points: Stability, eccentricity, and applications to mobile facility location. Int. J. Comput. Geom. Appl. 16, 345–372 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Durocher, S., Kirkpatrick, D.G.: Bounded-velocity approximation of mobile Euclidean 2-centres. Int. J. Comput. Geom. Appl. 18, 161–183 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Gaffney, S., Smyth, P.: Trajectory clustering with mixtures of regression models. In: Proc. 5th KDD, pp. 63–72 (1999)Google Scholar
  12. 12.
    Gudmundsson, J., van Kreveld, M., Speckmann, B.: Efficient detection of patterns in 2D trajectories of moving points. GeoInformatica 11, 195–215 (2007)CrossRefGoogle Scholar
  13. 13.
    Halperin, D.: Arrangements. In: Goodmann, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Comput. Geom, pp. 529–562. Chapman & Hall/CRC, Boca Raton (2004)Google Scholar
  14. 14.
    Har-Peled, S.: Taking a walk in a planar arrangement. SIAM J. Comput. 30(4), 1341–1367 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Hershberger, J., Snoeyink, J.: Computing minimum length paths of a given homotopy class. Comput. Geom. 4, 63–97 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Kedem, K., Livne, R., Pach, J., Sharir, M.: On the union of jordan regions and collision-free translational motion amidst polygonal obstacles. Discrete Comput. Geom. 1, 59–70 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Laube, P., Purves, R.S.: An approach to evaluating motion pattern detection techniques in spatio-temporal data. Comp., Env. and Urb. Syst. 30, 347–374 (2006)CrossRefGoogle Scholar
  18. 18.
    Lee, J., Han, J., Whang, K.-Y.: Trajectory clustering: a partition-and-group framework. In: Proc. ACM SIGMOD Int. Conf. Man. of Data, pp. 593–604 (2007)Google Scholar
  19. 19.
    Lee, J.-G., Han, J., Li, X., Gonzalez, H.: TraClass: Trajectory classification using hierarchical region-based and trajectory-based clustering. In: PVLDB 2008, pp. 1081–1094 (2008)Google Scholar
  20. 20.
    van der Stappen, A.F., Halperin, D., Overmars, M.H.: The complexity of the free space for a robot moving amidst fat obstacles. Comput. Geom. 3, 353–373 (1993)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Kevin Buchin
    • 1
  • Maike Buchin
    • 2
  • Marc van Kreveld
    • 2
  • Maarten Löffler
    • 3
  • Rodrigo I. Silveira
    • 4
  • Carola Wenk
    • 5
  • Lionov Wiratma
    • 2
  1. 1.Dept. of Mathematics and Computer ScienceTU Eindhoven 
  2. 2.Dept. of Information and Computing SciencesUtrecht University 
  3. 3.Dept. of Computer ScienceUniversity of CaliforniaIrvine
  4. 4.Dept. de Matemàtica Aplicada IIUniversitat Politècnica de Catalunya 
  5. 5.Dept. of Computer ScienceUniversity of Texas at San Antonio 

Personalised recommendations